That seems odd to me. A high accuracy initial guess seems to me an initial guess which is very near to the solution. But if you find such a thing, then you have the solution with a high accuracy. So why should you use Newton's method then?

I don't don't the solution yet , But I know the interval in which the solution lies .

Sorry for my English.

:)

My point is, it doesn't matter what point you choose. If you are lucky enough(which you are with a high probability because troubles are not very probable), that initial guess will be good enough to give you the solution with desirable accuracy in a good number of steps.
But if you want to have an educated guess, you can try to draw(or imagine!!!) the graph of the function and see where are the zeroes. This way you can have a very good guess but it doesn't garuantee that the steps will be fewer because sometimes you get far from the solution and reach it again. So just make a guess.

In general, I think there are no rules. For special cases, there may be methods to bound the roots to an interval. For polynomials, there are several rules to help locate the roots. See http://en.wikipedia.org/wiki/Properties_of_polynomial_roots This may also help if you have a polynomial approximation to the function you are working with.

There really isn't a ``rule" for which ##x##-value to pick for the intial ``guess" in Newton's Method. Just look at the function and use some analytic techniques (like FactChecker mentioned) to try to reason where the roots might be, then pick the nearest integer around there.

Be aware that Newton's method can diverge if the function has a bad shape. For those functions, it can take several tries to get an answer. So put a limit on the number of iterations and start again with a different initial guess when the limit is exceeded.